Angle Bisector

Angle Bisector in geometry is a line, ray, or segment that divides an angle into two equal angles of the same measure. The word Bisector means dividing a shape or an object into two equal parts. In the case of geometry, it is often used to split triangles and angles into equal measures.

In this article, we will discuss the introduction, definition, and properties of an Angle Bisector and its meaning. We will also understand the construction of an Angle Bisector and the theorem to calculate the angle. We will also solve various examples and provide practice questions for a better understanding of the concept of this article.

What is an Angle Bisector?

An Angle Bisector, within geometry, is a ray, line, or segment that effectively splits a given angle into two equal parts. In simpler terms, it’s a method of dividing an angle into two equal angles. For example, if you need to create a 60° angle, you can achieve this by first constructing a 120° angle and then dividing it using an angle bisector. Similarly, angles like 90 degrees, 45 degrees, and 15 degrees can also be constructed using the concept of an angle bisector. An example of an Angle Bisector can be observed in a clock. This occurs when the angle formed by the minute hand and the hour hand is divided equally by the presence of the second hand.

Angle Bisector Definition

An Angle Bisector can be defined as a ray or line segment that, when applied to a given angle, splits it into two angles of the same measure. In simpler terms, it is a way of dividing an angle into two congruent or equal angles.

Angle Bisector of Triangle

In a triangle, an Angle Bisector is a straight line that splits an angle into two equal or congruent parts. Ever triangle consists three vertices and three associated triangle. There can have up to three angle bisectors. Angle bisectors of a triangle can be classified into two types depending on the angles they bsect

• Internal Angle Bisector
• External Angle Bisector

Internal Angle Bisector

Internal Angle Bisector as the name suggest bisects the interior angles of a triangle. There are three such internal bisectors with one originating from each vertex. The meeting point of these three angle bisectors within a triangle is referred to as “incenter.” This incenter is equidistant from all three vertices of the triangle. AD is the internal angle bisector of angle A in triangle ABC.

External Angle Bisector

External Angle Bisector are the bisectors of exterior angles of a triangle. Exterior angles of a triangle can be formed by extending one of its sides. There can be three such exterior angles and hence corresponding to them there can be three angle bisectors. AE is external angle bisector of angle CAD in triangle ABC.

Properties of Angle Bisector

The properties of Angle Bisector are mentioned below:

• An Angle Bisector splits an angle into two angles of precisely the same measurement.
• All the point positioned along this angle bisector is equidistant from the both arms or sides of the angle.
• In a triangle, the angle bisector divides the side opposite to the angle in ratio that is equal to the ratio of the other two sides.

Construction of an Angle Bisector

To construct an Angle Bisector for an angle using a ruler and compass, follow these steps:

Step 1: Draw an angle say ∠PQR.

Step 2: Put the pointer of compass at point “Q.” Proceed to draw an arc that intersects the two angle arms, “PQ” and “QR,” at different points.

Step 3: Afterward, reposition the compass pointer at the intersection of the first arc with line “PQ.” Create another arc inside the angle.

Step 4: Without altering the compass radius, move the pointer to where the second arc crosses the arm “QR.” Draw an additional arc within the angle, ensuring it intersects with the prior one.

Step 5: Finally, by using a ruler to draw a straight line from point “Q” to the intersection point of the two interior arcs within the angle.

By following these steps, we can successfully construct an angle bisector using a compass and ruler.

Angle Bisector Theorem

The Angle Bisector Theorem states that within a triangle, when an angle bisector is drawn from a vertex and it extends to one side, this bisector divides that side in a proportion equivalent to the ratio of the other two sides of the triangle, it means that the relationship between the segments formed by the angle bisector on one side is directly linked to the relative lengths of the other two sides of the triangle.

Interior Angle Bisector Theorem

The Interior Angle Bisector Theorem states that the interior angle bisector of a triangle joining the opposite side divide the opposite sides in proportion to the remaining two sides of the triangle.

External Angle Bisector Theorem

External Angler Bisector Theorem states that the bisector of exterior angle of a triangle divides the opposite side externally in the ratio equal to the ratio of the sides enclosing the angle.

Perpendicular Bisector Theorem

The Perpendicular Bisector Theorem states that if a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. In other words, if a point is equidistant from the endpoints of a line segment, it must lie on the perpendicular bisector of that line segment.

It is often used in geometry to show that a point is on the perpendicular bisector of a line segment or to prove properties of triangles. When dealing with triangles, if you have a point on the perpendicular bisector of one of the sides, it will divide that side into two congruent segments.

Solved Example of Angle Bisector

Example 1: If an Angle Bisector divides an angle of 120 degrees, then what will be the measure of each angle?

Solution:​

Given, a angle is 120 degrees

As we know, the Angle Bisector splits the angle into equal two parts.

Therefore, 120 degrees is divided into equal two parts, say z.

Hence,

z + z = 120°

2z = 120°

z = 120°/2

z= 60°

Example 2: A ray PX, divides an angle PQR into two equal parts. If one part is equal to 5x – 5 and the second part is equal to 20, then what is the value of x?

Solution:

Given, PX divides angle PQR into two equal parts. Thus, PX is the angle bisector.

Now, each part should measure equal.

Thus,

5x – 5= 20

5x = 20 + 5 = 25

x = 25/5 = 5

Hence, the value of x is 5.

Example 3: A ray QT is the bisector of ∠PQR and QU bisects ∠PQT. Find the measure of ∠TQU given that ∠PQR=120.

Solution:

It is given that ∠PQR=120°.
Also, ∠PQT = 1/2 × ∠PQR = 1/2 × 120° = 60° (QT is an angle bisector bisecting ∠PQR into two equal parts)
Now, ∠TQU = 1/2 × ∠PQT = 1/2 × 60° = 30° (QU is a bisector and bisects ∠PQT into two equal parts)
∴ The value of ∠TQU is 30°.

Example 4: A ray drawn from point O is the Angle Bisector of ∠POQ. If both bisector angle are 7x-2 and 12 then find the value of x.

Solution:

To find x, we will use that any point on the bisector of an angle is equal distance from the sides of the angle.
So, the bisector drawn from O will be same distance from sides OP and OQ
⇒ 7x − 2 = 12
⇒ 7x = 2 + 12
⇒ 7x = 14
∴ The value of x is 2.

Example 5: If ∠ACF = ∠FCB = 60° and CF is the angle bisector, find ∠ACB.

Solution:

We know that an Angle Bisector splits an angle into two equal sections.

Since ∠ACF = ∠FCB = 60°, we can say that ∠ACB= 60°+ 60°= 120°.

Practice Questions on Angle Bisector

Q1. If an Angle Bisector divides 280degrees, then what will be the measure of each angle?

Q2. Construct the Angle Bisector of 100 degrees angle.

Q3. Construct an Angle Bisector of 20 degrees angle.

Q4. If an Angle Bisector splits a right angle, what will be the measure each angle?

Q5. Find the value of x if the ray OM is an Angle Bisector and both angle is 3x-8 and x.

Q6. If an Angle Bisector splits an angle , what would be the measurement of the each angles thus formed?

Angle Bisector – FAQs

1. What is Meaning of Angle Bisector?

An Angle Bisector in geometry is a line, ray, or segment that divides an angle into two equal angles of the same measure.

2. What are the Properties an Angle Bisector?

The properties of Angle Bisector are:

An Angle Bisector splits an angle into two angles of precisely the same measurement.

All the point positioned along this angle bisector is equidistant from the both arms or sides of the angle.

In a triangle, the Angle Bisector divides the side opposite to the angle in ratio that is equal to the ratio of the other two sides.

3. Can an Angle Bisector divide the Sides of a Triangle?

In a triangle, the Angle Bisector divides the side opposite to the angle in ratio that is equal to the ratio of the other two sides.

4. How many Angle Bisector can Bisects an Angle?

There can be only one Angle Bisector to an angle.

5. What is an Angle Bisector of a Triangle?

In a triangle, an Angle Bisector is a straight line that splits an angle into two equal or congruent parts. Ever triangle consists three vertices and three associated triangle. There can have up to three angle bisectors, with one originating from each vertex. The meeting point of these three angle bisectors within a triangle is referred to as “incenter.” This incenter is equidistant from all three vertices of the triangle.

6. Does Angle Bisector Cut an Angle in Half?

Yes, an Angle Bisector divides the given angle into two similar angles. In other way, we can say that the measurement of each of angles is half of the original angle.

7. How to Construct an Angle Bisector?

To construct an Angle Bisector for an angle using a ruler and compass, follow these steps:

Step 1. Draw an angle say ∠XYZ.

Step 2. Put the pointer of compass at point “Y.” Proceed to draw an arc that intersects the two angle arms, “XY” and “YZ,” at different points.

Step 3. Afterward, reposition the compass pointer at the intersection of the first arc with line “XY.” Create another arc inside the angle.

Step 4. Without altering the compass radius, move the pointer to where the second arc crosses the arm “YZ.” Draw an additional arc within the angle, ensuring it intersects with the prior one.

Step 5. Finally, by using a ruler to draw a straight line from point “Y” to the intersection point of the two interior arcs within the angle.

8. Can the Angle Bisector pass through the Midpoint?

The Angle Bisector does not always pass through the midpoint of the opposite side of a triangle. Instead, it divides the opposite side in a manner that is directly related to the lengths of the adjacent sides of the triangle.

9. How many Angle Bisector does a Triangle have?

A triangle has three angles, it can possess only three angle bisectors.

10. What is Incenter?

An incenter is the point of intersection where all three angle bisectors of a triangle converge.

11. What is the importance of an Angle Bisector?

Angle Bisector serve a vital role in geometry by facilitating the identification of corresponding parts in similar triangles. They help to establish a proofs and relationships concerning these triangles.

12. How to Prove Angle Bisector Theorem?

To prove the Angle Bisector Theorem, follow these steps:

1. Draw a triangle with an angle bisector.
2. Apply the Angle Bisector Theorem, which states that the ratio of the lengths of the two segments created by the angle bisector is equal to the ratio of the lengths of the two opposite sides.
3. Use this theorem to prove the desired relationship between the sides.